Originally Posted by Throckmorton
Bernoulli's Principle isn't why planes fly. They fly by redirecting airflow downward. That's why pitch is used to adjust the amount of lift the wings produce.The FAA tests ask a pilot how a wing generates lift, and the pilot has to answer "Bernoulli's Principle". Then when he flies the plane, he modulates lift by angling the wings and flaps with no regard for Bernoulli and his principle. Interesting isn't it?
There is one thing I don't understand though. Every explanation says that Bernoulli accounts for a small portion of lift, then they also say that according to the equations it generates the same lift as redirecting airflow. How can both be true? What exactly are the equations based on? Are they set up specially to match observations?
You didn't read my post at all did you
Well, that nasa page has the story exactly right. Didn't read the other links. There are "two ways" to explain lift:
1) Newton's 3rd law (as you indicated, the lifting body changes the direction of airflow; it must have exerted a force on the air to do this)
2) Pressure on the upper wing suface is lower than presure on the lower wing surface. Net upward pressure = lift
In reality these things are one and the same. Let me explain in more detail...
The Navier-Stokes Equations are an excellent model for fluid flow on a very large scale. Any fluid is made up of a huge number of molecules. Tracking the behavior of every molecule individually is unbelievably expensive if we're talking about a region of space that's more than a few microns on each side. But if we're looking at an amount of fluid in say the earth's atmosphere, then it turns out that the microscopic behavior of every molecule doesn't really matter. So we can assume that the fluid is a continuum--instead of being made of a whole bunch of tiny particles, it's one continous blob.
Using that assumption, Newton's laws, and some calculus (well, maybe a lot of calculus), you can derive the Navier-Stokes Equations. In 3D, they are a set of coupled partial differential equations describing the conservation of mass, momentum, and energy. These do a great job describing fluid flow; so much so that the enginerring and scientific community accept them basically as law in the macroscopic setting. Unfortunately, solving the Navier-Stokes equations exactly is an unsolved problem except in some very, very special cases. But we can approximate them reasonably well using (big) computers.
The Euler Equations arise from setting viscosity = 0 in the Navier-Stokes Equations. It's a HUGE simplification but Euler Eqns are still very hard to solve. They govern inviscid flow at the most general level. As I mention in my previous post, the Bernoulli equation is a special case of the euler equations. Bernoulli results from considering the euler equations along a single streamline in incompressible (density=constant; reasonable for most liquids and gasses moving at low speed), steady (not time varying) flow. You can only make comparisons across multiple streamlines if the flow is also irrotational (i'm not going to explain this one but suffice it to say, most aerodynamic flows are). When you have all of these conditions, the flow is called "potential" (same thing mathematically as electric potential fields if you're familiar with that). So that should answer your question about where Bernoulli comes from.
At its core, Bernoulli describes how changes in fluid velocity affect changes in fluid pressure.
Anyway, even without big computers, I can apply the Navier Stokes or Euler equations to make interesting qualitative statements about fluid mechanics. For example, imagine an isolated airfoil in "flight". For simplicity, the fluid is inviscid so drag=0. You observed that it feels a lifting force b/c it deflects the air downard. Ok. Now imagine drawing a box surrounding the airfoil*. Say I measure the momentum of the air crossing each face of my box. If I add up (read: integrate) all the momenta, I'll notice that the momentum changed! Why? B/c the airfoil exerted a force on the air. So this change in momentum will be precisely the lift. That was very qualitative. I can make it a little more quantitative by attempting to evaluate all the terms of the Euler Equations. What I will find is that depending on how I draw the box, I could find that:
Lift1 = change in momentum of air; 1) from above
Lift2 = pressure difference of the top/bottom surface of my airfoil; 2) from above
Lift3 = the average of the above two quantities
Note that the value of Lift is fixed, so 1) and 2) produce the same results, Lift1=Lift2. So Lift3=(Lift1+Lift2)/2 is the same thing too. So momentum or pressure, or both are valid ways of explaining lift. It all depends on your frame of reference.
*Ok I'm cutting corners here. Technically the box cannot contain the airfoil. Mathematically you can get around this issue easily but it's hard to explain without a diagram & hard to explain if you aren't familiar with path integrals. So I'll skip it for now...
SO it's not really that Bernoulli accounts for lift. If I knew the exact velocity distribution around an aircraft in *inviscid* *incompressible* *steady* *irrotational* flow, then I could use Bernoulli (which relates velocity to pressure) to compute what the pressure distribution is. From that, I could compute the lift.
In a viscous flow, if I knew the exact velocity distribution around the "extended body" (described in my previous post), I could use Bernoulli to *approximate* the lift. (This turns out to be a good approximation).
However, in both cases, knowing the exact velocity distribution is really hard. So hard that nobody attempts to compute lift by measuring those things physically. But it often is a reasonable way to compute lift on the computer, where you have solved the euler equations (approximately) and have access to the approximate velocity values at any location.
The issue is that people twist Bernoulli. They say silly things like air travels farther over the top of the airfoil than the bottom, so it has to move faster over the top (patently false reasoning for a true statement). Since it's moving faster, the pressure is lower ("true" by Bernoulli). Hence lift is generated. ("True" in quotes b/c you have to be careful where/how you apply Bernoulli in real physical flows, since they aren't incompressible, steady, invscid, and irrotational. With the right assumptions you can get a reasonable approximation.)
The right reasoning is: we observe that lift is generated; one valid explanation is pressure difference. Bernoulli then tells us that hte air is moving faster on the side with lower pressure than the side with higher pressure. If you knew the pressure at every point, you could use Bernoulli to recover the velocity values. But this has absolutely NOTHING to do with the top surface being "longer" or any crazy crap like that.